同时optimization、优控和（双）次模多面体的特征
A characterization of simultaneous optimization, majorization, and (bi)submodular polyhedra
摘要：
出于电源管理应用中的资源分配问题（RAP），我们调查的解决方案，同时最小化整个类的目标函数的最优化问题。这是直接的经验表明，这样的解决方案不存在大多数最优化问题。然而，很少有人知道为什么会出现这种情况，以及是否存在这样的解决方案的问题的特征。在这篇文章中，我们回答了这些问题的解决方案，同时优化类舒尔凸函数，称为最少优元，（双）次模函数和相应的多面体的存在性。为此，我们引入了一个广义的优化和最少优化元素，称为$(a,b)$ -优化和最少$(a,b)$ -优化元素，并描述了可行的问题集，这些元素在这些多面体。据此，我们还获得了新的基础和双次模多面体，从20世纪70年代的线性最优化的最佳贪婪算法方面，这些集的经典特征扩展。我们讨论了我们的研究结果在电源管理应用中的RAP的影响，并使用结果来获得一个新的表征的凸合作对策和新的性能的最佳估计的特定正则化回归问题。在一般情况下，我们的研究结果突出了同时优化解决方案的组合性质，并在同一时间，提供了一个理论解释的观察，这样的解决方案一般不存在。
英文摘要：
Motivated by resource allocation problems (RAPs) in power management applications, we investigate solutions to optimization problems that simultaneously minimize an entire class of objective functions. It is straightforward to show empirically that such solutions do not exist for most optimization problems. However, little is known on why this is the case and whether a characterization exists of problems that do have such solutions. In this article, we answer these questions by linking the existence of solutions that simultaneously optimize the class of Schur-convex functions, called least majorized elements, to (bi)submodular functions and the corresponding polyhedra. For this, we introduce a generalization of majorization and least majorized elements, called $(a,b)$-majorization and least $(a,b)$-majorized elements, and characterize the feasible sets of problems that have such elements in terms of these polyhedra. Hereby, we also obtain new characterizations of base and bisubmodular polyhedra that extend classical characterizations of these sets in terms of optimal greedy algorithms for linear optimization from the 1970s. We discuss the implications of our results for RAPs in power management applications and use the results to derive a new characterization of convex cooperative games and new properties of optimal estimators of specific regularized regression problems. In general, our results highlight the combinatorial nature of simultaneously optimizing solutions and, at the same time, provide a theoretical explanation for the observation that such solutions generally do not exist.
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